A Lagrangian Study of Precipitation-Driven Downdrafts*

GIUSEPPE TORRI

Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

ZHIMING KUANG

Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences,

Harvard University, Cambridge, Massachusetts

(Manuscript received 29 July 2015, in final form 9 November 2015)

ABSTRACT

Precipitation-driven downdrafts are an important component of deep convective systems. They stabilize

the atmosphere by injecting relatively cold and dry air into the boundary layer. They have also been invoked

as responsible for balancing surface latent and sensible heat fluxes in the heat and moisture budget of tropical

boundary layers. This study is focused on precipitation-driven downdrafts and basic aspects of their dynamics

in a case of radiative–convective equilibrium. Using Lagrangian particle tracking, it is shown that such

downdrafts have very low initial heights, with most parcels originating within 1.5 km from the surface. The

tracking is also used to compute the contribution of downdrafts to the flux of moist static energy at the top of

the boundary layer, and it is found that this is on the same order of magnitude as the contribution due to

convective updrafts, but much smaller than that due to turbulent mixing across the boundary layer top in the

environment. Furthermore, considering the mechanisms driving the downdrafts, it is shown that the work

done by rain evaporation is less than half that done by condensate loading.

1. Introduction

The existence of downdrafts in deep convective sys-

tems was acknowledged as early as a hundred years ago

by Humphreys (1914), who also hypothesized that they

are maintained by the evaporation of rain falling

through unsaturated air. Since then, downdrafts have

been the focus of many studies (e.g., Knupp and Cotton

1985; Cotton et al. 2011, and references therein), which

have led to a considerable advancement of our un-

derstanding of these phenomena and, more generally, of

precipitating convection.

Within the category of downdrafts, we would like to

concentrate on precipitation-driven downdrafts, bywhich,

following the nomenclature explained in Knupp and

Cotton (1985), wemean those downdrafts that appear in

the presence of precipitation and that reach the surface,

injecting masses of cold and dry air in the subcloud layer

(e.g., Byers and Braham 1949; Fankhauser 1976; Barnes

and Garstang 1982; Knupp 1987, 1988).

From a rather practical point of view, precipitation-

driven downdrafts have been of interest for some time,

given their potential to generate downbursts and mi-

crobursts, very strong currents of air with vertical ve-

locities often exceeding 10m s21 in magnitude (see, e.g.,

Fujita 1985, 1986; Proctor 1988, 1989; Wakimoto et al.

1994). These intense downdrafts can be found in cases

where precipitating clouds are formed above deep and

dry boundary layers—such as over the Great Plains—

and constitute a serious hazard for civil aviation.

From a more conceptual perspective, throughout the

years precipitation-driven downdrafts have been studied

in different contexts, an important example being that of

the so-called boundary layer quasi-equilibrium (BLQE)

hypothesis (Emanuel 1989; Raymond 1994, 1995), pro-

posed to explain why the moist entropy of tropical

boundary layers seems to change very little with time.

Simply put, according to this hypothesis, the moistening

and heating of the tropical boundary layer provided by

* Supplemental information related to this paper is available at

the Journals Online website: http://dx.doi.org/10.1175/JAS-D-15-

0222.s1.

Corresponding author address: Giuseppe Torri, Department of

Earth and Planetary Sciences, Harvard University, 20 Oxford

Street, Cambridge, MA 02138.

E-mail: torri@fas.harvard.edu

FEBRUARY 2016 TORR I AND KUANG 839

DOI: 10.1175/JAS-D-15-0222.1

� 2016 American Meteorological Society

the surface fluxes are counterbalanced mainly by

precipitation-driven downdrafts that bring low-entropy

air from the midtroposphere to the surface. As argued by

Raymond (1995), the turbulent entrainment at the top of

the boundary layer plays little role in the balance. This

explanation is in contrast with the hypothesis used by

Arakawa and Schubert (1974) in their cumulus parame-

terization, where it is mixing with the environmental air

and, sometimes, horizontal advection, which balance the

action of surface fluxes. These two hypotheses were re-

cently examined and tested by Thayer-Calder and

Randall (2015) with a numerical model. Interestingly,

their results suggest that only a small fraction of the flux

of moist static energy (MSE) at the top of the boundary

layer is due to precipitation-driven downdrafts.

Another framework in which precipitation-driven

downdrafts are of considerable importance is that of

convection triggering. The masses of air injected by

downdrafts in the subcloud layer spread horizontally on

the surface as density currents, usually called cold pools,

which are one of the main factors responsible for

maintaining deep convective systems (Purdom 1976;

Weaver and Nelson 1982). In organized cases, such as

squall lines or supercell thunderstorms, it has been

known for a long time (see, e.g., Rotunno and Klemp

1985; Rotunno et al. 1988; Weisman et al. 1988;

Weisman and Rotunno 2004) that the interaction be-

tween the gust front of cold pools and the wind shear or

the collision between two or more cold pools can pro-

vide enough kinetic energy to parcels at the surface to

reach their level of free convection (LFC). In non-

organized cases, parcels reach their LFC through a co-

operation of the gust-front lifting and the thermodynamic

effect provided by the positive moisture anomaly at the

leading edge of cold pools, which reduces the convective

inhibition experienced by parcels above the lifting con-

densation level (Tompkins 2001; Torri et al. 2015). The

influence of precipitation-driven downdrafts on convec-

tion triggering is indirect but, nevertheless, strong: by

controlling the initial conditions of cold pools, downdrafts

effectively control the strength of the cold pools’ gust

fronts and their ability to lift parcels from the surface

and, potentially, also the magnitude of the thermo-

dynamic effect.

In this study, we will consider a tropical maritime

environment, and we will focus on three fundamental

aspects of precipitation-driven downdraft dynamics:

1) What is the initial height of a precipitation-driven

downdraft?

2) What is the contribution of precipitation-driven

downdrafts to the budget of MSE in the subcloud

layer?

3) What is the role of condensate loading versus rain

evaporation in maintaining a precipitation-driven

downdraft?

To some extent, these topics have been addressed in

the past in various studies, although using limited tech-

niques that did not provide an entirely clear answer. For

instance, starting from as early as Mal and Desai (1938),

the typical approach to diagnose the initial height of

downdrafts, sometimes referred to as source level, has

been to use thermodynamic properties of downdraft air,

like the wet-bulb potential temperature uw, the equiva-

lent potential temperature ue, or MSE, which are ap-

proximately conserved in adiabatic processes and water

vapor–liquid water phase transitions (see also, e.g., Mal

and Desai 1938; Normand 1946; Newton 1950; Zipser

1969; Betts 1976; Fankhauser 1976; Lemon 1976; Barnes

and Garstang 1982; Johnson and Nicholls 1983;

Kingsmill and Houze 1999). The problem with this ap-

proach is that it assumes that mixing is negligible and

that all the downdraft air originates at the initial height,

neither of which hypotheses are necessarily true.

Analogously, various authors have asked what

mechanisms maintain precipitation-driven downdrafts,

recognizing rain evaporation and condensate loading as

leading factors (Byers and Braham 1949; Braham 1952;

Fankhauser 1976; Lemon 1976; Barnes and Garstang

1982; Knupp and Cotton 1985). However, the extent to

which one mechanism is dominant over the other even in

simple cases of unorganized convection remains unclear.

In this manuscript, we will address the two questions

outlined above mainly using a Lagrangian particle dis-

persion model (LPDM). Compared to the techniques

that have been used in the past, the LPDM provides a

great deal of information. In particular, analyzing the

position of millions of particles released in the simula-

tion domain will allow us to determine the initial height

of a downdraft in a clear and simple manner. Also,

combining the output from the LES model that we will

use with the Lagrangian history of the particles from

when they enter a precipitation-driven downdraft until

their exit near the surface, we will be able to quantify the

importance of rain evaporation and condensate loading

in driving the downdraft.

Notice that the use of particles’ trajectories to study

downdrafts is a path that has been explored in the past:

Miller and Betts (1977), for example, used them to il-

lustrate the behavior of downdrafts in Venezuelan

convective storms, and Knupp (1988) used them in the

context of storms in the high plains. However, the

computational limitations of the past did not allow for

the simulation of a number of trajectories large enough

to make significant statistics.

840 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

2. Methods

The model used in this study is the System for At-

mospheric Modeling (SAM), version 6.8.2, which solves

the anelastic equations of motion and uses liquid water

static energy and nonprecipitating and precipitating

total water as thermodynamic prognostic variables

(Khairoutdinov and Randall 2003). The equations are

solved with doubly periodic boundary conditions in the

horizontal directions. A prognostic turbulent kinetic

energy 1.5-order closure scheme is used to parameterize

subgrid-scale effects.

We ran the model with the Lin microphysics scheme,

a single-moment scheme that predicts the specific hu-

midities of water vapor, cloud liquid water, cloud ice,

rain, snow, and graupel (Lin et al. 1983). The surface

fluxes are computed using the Monin–Obukhov simi-

larity theory.

The simulations are carried out over a domain of 64364km2 in the horizontal directions and 30 km in the

vertical. The grid size in the horizontal directions is

250m, and it varies in the vertical: 500m above 10 km,

which decrease progressively to 38m close to the model

surface. The time step is 3 s. Figure 1 shows the envi-

ronment vertical profiles of potential temperature

(red), water vapor specific humidity (blue), and relative

humidity (green), averaged over the duration of the

simulation.

We ran the model with the LPDM discussed in Nie

and Kuang (2012). For every column in the domain, the

LPDM is initialized with 1120 particles, their positions

being distributed randomly over the bottom 18kmof the

model domain, with a probability distribution that is a

uniform function of pressure. A validation of the LPDM

can be found in the supplemental information.

Every minute of simulated time, the positions of the

Lagrangian particles and the three-dimensional model fields

fromSAMare recorded.A cross-comparison of Lagrangian

and Eulerian data provides the complete histories of ther-

modynamic and dynamic properties of the particles.

Throughout the simulation, the sea surface tempera-

ture is held constant at 302.65K, and the diurnal cycle is

removed by fixing the zenith angle at 51.78 with solar

constant halved to 685Wm22. The mean winds are ini-

tialized to zero. We first ran the model for 30 days

without Lagrangian particles on the same domain until

radiative–convective equilibrium (RCE) is reached. At

the end of this period, we restarted the simulation with

the Lagrangian particles and collected data for 12 h of

model time. No nudging was imposed on the horizontal

winds. The precipitation rate averaged in time during

the 12h of data collection and in space over the entire

domain is 3.4mmday21.

We define the subcloud layer as the portion of the

model domain that includes the surface and in which

the time- and domain-averaged cloud liquid water

specific humidity ql is below 1025 g kg21. With the

chosen settings, this corresponds to the bottom 531m

of the domain.

We define the density potential temperature of a

particle as follows (Emanuel 1994):

ur5 u

�11

�R

y

Rd

2 1

�qy2 q

l

�, (1)

where Rd and Ry are the gas constants of dry air and

water vapor and equal 287 and 461 J kg21K21, re-

spectively, and qy is the specific humidity of water vapor.

As customary, we define «5 (Ry/Rd 2 1) and will use

this convention for the remainder of this manuscript.

We say that a grid box in the subcloud layer is part of

the core of a cold pool if its density potential

FIG. 1. Time average of the vertical profiles of potential tem-

perature (red), water vapor specific humidity (blue), and relative

humidity (green).

FEBRUARY 2016 TORR I AND KUANG 841

temperature ur is lower than the horizontal average by

1.5K.1 This definition was chosen after examining the

model outputs and only serves to separate the part of the

cold pool that is fed by the precipitation-driven down-

draft from the rest (cf. Torri et al. 2015). As shown in the

supplemental information, the analysis presented in the

following section is not particularly sensitive to the value

chosen for the threshold.

We consider as part of a precipitation-driven downdraft

all particles from any altitude that acquire and maintain

negative vertical velocity until they enter the core of a

cold pool. Interestingly, although we do not require the

precipitation-driven downdraft to be unsaturated, most of

the descent of the particles in precipitation-driven down-

drafts takes place in unsaturated grid boxes.

3. Results

Having laid out the research questions in section 1,

and after having explained how we pursue these ques-

tions in the section 2, we will now discuss our findings.

a. On the initial height of precipitation-drivendowndrafts

Following the definition of precipitation-driven down-

draft that we have given in the previous section,

whenever a particle enters the core of a cold pool, we

track its position back in time for every step during which

the particle was in grid boxes that satisfied the condition

that the vertical wind w# 0ms21. The height of the first

grid box for which the previous condition is true is taken

as the initial height of the particle we sampled.

Notice that, because the coldest regions of cold pools

are known to exhibit high pressure perturbations, the

model outputs show that the vertical velocity of

precipitation-driven downdrafts as they approach the

surface decreases considerably (cf. Fig. 10 later in the

text). When the vertical velocity is small enough, tur-

bulent or wave fluctuations may cause its values in some

grid boxes to oscillate between positive and negative.

There follows that, if we were to apply the definition of

precipitation-driven downdraft given above, a consid-

erable number of sampled particles would appear to

have originated close to the surface. To avoid this

problem, whenever the vertical velocity of a particle in a

downdraft turns positive, a clock is started as the parti-

cles’ positions are scanned backward in time; if the

vertical velocity becomes negative again within 10min

from the start of the clock, the particle is still considered

part of the downdraft; otherwise, it is dismissed. Sensi-

tivity tests presented in the supplemental information

show that the results and our conclusions are not par-

ticularly sensitive to the precise value chosen for the

time threshold. Furthermore, in order to avoid in-

terference from fast detrainment–entrainment events in

cold pools, once a particle ends up in a cold pool core

and its properties are recorded, the particle is flagged

and discarded for the rest of the simulation.

In Fig. 2, the blue curve shows the distribution of initial

heights of all particles that were injected in the core of a

cold pool by a precipitation-driven downdraft. Notice that

the distribution is confined almost entirely to the bottomof

the troposphere: 98%of particles have initial heights lower

than 2.5km. The curve shows a pronounced peak at ap-

proximately 1km of altitude, just a few hundred meters

above the top of the subcloud layer, and a smaller one near

the surface. Examination of trajectories for particles

that contribute to the lower peak shows that most of

these particles descended from higher altitudes with a

precipitation-driven downdraft, but, near the surface, their

vertical velocity, however small, was positive formore than

10min before they entered the core of a cold pool.

These results are nicely in accordance with existing

literature, which suggested that, for tropical convection,

the source level of precipitation-driven downdrafts is just

FIG. 2. Distribution of Lagrangian particles’ positions at various

times before becoming part of a precipitation-driven downdraft.

The red, green, and black curves refer to positions 30, 60, and

90min before entering the downdraft, respectively. The blue curve

shows the distribution of positions at the time of entrance.

1 For brevity, whenever a particle enters a grid box that is part of

the core of a cold pool, we will simply say that ‘‘the particle enters/

belongs to the core of a cold pool.’’

842 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

above cloud base [see, e.g., Zipser (1969), Betts (1976),

and Kingsmill and Houze (1999), just to cite a few].

We should point out that the initial height of a

downdraft may still carry along some ambiguity. For

example, it could be that a number of particles that

entered the precipitation-driven downdraft at 1 km ac-

tually originated frommuch greater altitudes: they were

transported down by another downdraft from which

they eventually left, and then, after some time, they

became part of the precipitation-driven downdraft that

carried them down to the surface. A simple way to check

whether this is the case is to ask where the particles were

before entering the precipitation-driven downdraft.

The red, green, and black curves in Fig. 2 show the

distribution of the sampled particles’ positions 30, 60,

and 90min before entering the precipitation-driven

downdraft, respectively. They can be thought of as the

evolution back in time of the blue curve. Interestingly,

the red curve suggests that 61% of particles actually

come from the subcloud layer, whereas the remaining

seem to originate near the height where they entered the

downdraft. According to the nomenclature discussed in

Knupp (1988), we will refer to the former as up–down

and to the latter asmidlevel. Less than 2%of all particles

originate from an altitude greater than 2.5 km.

To better appreciate the history of the Lagrangian par-

ticles prior to their entrance in a precipitation-driven

downdraft, we can look directly at the trajectories of a few

representative cases. The red lines in Fig. 3 show the time

evolution of the vertical positions of three particles for the

60min before they enter the core of a cold pool. Notice that

the horizontal axis represents the time in the particles’ his-

tories, with 0min being themoment they enter a cold pool’s

core. The particles have been selected rather arbitrarily, but

an extensive scan through a large number of particles’ his-

tories makes us confident the particular examples shown

illustrate well the possible scenarios for downdrafts.

In all panels, the blue contours represent the total

nonprecipitating water specific humidity qn, whereas the

green contours represent the buoyancy. To keep the image

clean, we only show contours for negative buoyancy and

only for the subcloud layer. Finally, the gray shading rep-

resents the total precipitating water specific humidity qp.

Figure 3 (left) shows the history of a particle in an up–

down downdraft. As can be seen, the particle is lifted from

the surface at approximately225min by the gust front of

an expanding cold pool, indicated by the appearance of

the negative buoyancy contours in the subcloud layer, and

it quickly becomes part of an updraft. As the particle as-

cends, it reaches lifting condensation level (LCL), and

soon afterward it enters a precipitating column. The pre-

cipitation experienced by the particle is sufficient to

change the sign of its buoyancy and stop its ascension.As a

result, the particle leaves the updraft and enters a down-

draft that becomes unsaturated almost immediately. The

descent continues with no interruptions until the particle

reaches the core of a cold pool. Curiously, visual in-

spection of the three-dimensional model output suggests

that the particle enters the same cold pool that had lifted it

from the surface.

FIG. 3. Three examples of Lagrangian particles’ histories 60min before entering the core of a cold pool are shown with a red line. (left)

An up–down downdraft, (middle) a midlevel downdraft originating at ;1 km, and (right) a hybrid between the two. The blue contours

represent the specific humidity of total nonprecipitating water at increasing intervals of 1 g kg21. The green contours show negative

buoyancy in the subcloud layer at decreasing intervals of 0.02m s22 starting at 20.01m s22. The gray shading represents the specific

humidity of total precipitating water. The values shown for all variables at different heights refer to the vertical column where the

particle is.

FEBRUARY 2016 TORR I AND KUANG 843

Figure 3 (middle) shows the history of a particle that

descends to the surface levels through a midlevel

downdraft. As might be expected, the particle sits in an

almost undisturbed environment for most of its life. The

changes in altitude in clear sky seem to be due to ran-

dom turbulent fluctuations or gravity waves. Then, at

approximately 220min, the particle is entrained into a

cloud, where its buoyancy becomes negative. The par-

ticle starts descending and enters the core of a cold pool

at the surface.

Finally, Fig. 3 (right) shows another possible scenario

for particles entering precipitation-driven downdrafts—

one that, to some extent, is a mixture of the previous two.

The particle represented in the figure starts off at an al-

titude of approximately 1km, subject only to random

turbulence or wave motions, which cause its position to

oscillate a little. At220min, the particle becomes part of

an updraft that makes it rise to an altitude of 2km, where,

finally, its buoyancy reverses sign and the particle enters

an unsaturated downdraft all theway down to the surface.

This strengthens the conclusion drawn from Fig. 2 that

particles with higher initial height may actually originate

atmuch lower altitude and have, therefore, a higherMSE

than what the initial height could suggest.

One of the messages that emerges from the previous

discussion is that precipitation-driven downdrafts tend to

have relatively low initial heights, possibly transporting

parcels that originated in the bottom kilometer of the

troposphere. To provide further evidence to support this

statement, we will use three methods, one based on the

Lagrangian and two on the Eulerian point of view.

The former one consists of changing the sampling

technique we have adopted above, and, instead of con-

sidering particles that enter a downdraft and then reach

the surface layers inside the core of a cold pool, we

simply track all particles in downdrafts, and we follow

them until they leave it, regardless of the height where

this happens. To be more specific, whenever a particle

enters a grid box in which w # 20.5m s21, its positions

are saved for as long as the particle is in a grid box with

negative vertical velocity (i.e., w # 0ms21).

For every particle, we collect the initial height and the

final height, which, in contrast to the previous sampling

of precipitation-driven downdrafts, does not necessarily

have to be in a cold pool or even in the subcloud layer.

We also do not allow for a temporal window between

the time a particle leaves the downdraft and when its

final height is recorded, as done with the other sam-

pling technique. Nevertheless, in the supplemental in-

formation we show that considering such a window has

no effect on our conclusions.

Next, we define the height matrix as the matrix where

the (i, j) element contains the density of particles per

meter square that entered a downdraft at height zj and

left it at height zi. Figure 4 shows the logarithm of this

matrix. As a guiding rule, numbers in the same column

refer to particles that became part of a downdraft at the

same height.

The figure suggests that, although there are down-

drafts at all heights, they do not travel much farther

than a few hundred meters. The reason for this, as also

suggested by others (see, e.g., Böing et al. 2014), is that,

as downdraft parcels descend over large distances, they

become dry, and, thus, they start following a dry adiabat.

Because the environment above the subcloud layer re-

mains close to a moist adiabat, these parcels quickly lose

their negative buoyancy. Furthermore, because pre-

cipitating columns are not steady, the negative contri-

butions of rain evaporation and condensate loading to

the parcels’ buoyancy are also going to become in-

sufficient to maintain the parcels’ descent.

One of the most curious features of Fig. 4 that jumps

out at the viewer is that downdrafts originating between

3 and 5km seem to be able to travel a greater distance

than those originating at other heights. The average

profile of rain evaporation rate shown by the blue curve

in Fig. 5, and that of the graupel melting rate shown in

green, suggests that particles in downdrafts that cross

the levels between 2 and 4km would experience signif-

icant evaporation of rain and melting of graupel, which

could provide extra kinetic energy to travel a longer

FIG. 4. Logarithm of the height matrix, the (i, j) entry of which

indicates the number of particles that became part of a downdraft

at height zj and exited at height zi.

844 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

distance than particles that descended frommuch higher

or lower altitudes. Notice that, although the peak in

melting rate of graupel is almost an order of magnitude

higher than the rain evaporation rate at the same alti-

tude, the latent heat of fusion is roughly a factor of 8

smaller than the latent heat of vaporization, so melting

of graupel is less efficient in cooling the parcels, and thus

in driving the downdraft, than rain evaporation. Finally,

we should note that the minimum of evaporation rate

between 1 and 2km is the result of high relative hu-

midity, and thus low saturation deficit, at these altitudes.

The second consistency check that we perform is

based on the distribution of MSE in cold pool cores, by

which, we remind the reader, we mean those grid cells

with a ur anomaly smaller than 21.5K. Because adia-

batic processes and phase transitions leaveMSE roughly

unchanged, and because MSE is monotonically de-

creasing in approximately the bottom 5km of the tro-

posphere, the presence of low-MSE grid boxes in cold

pool cores would be the signature of air parcels origi-

nating in the midtroposphere. The red curve in Fig. 6

shows the cumulative distribution function of MSE in

cold pool cores during the entire simulation, whereas the

blue curve represents the temporal and spatial average

of the environmental profile of MSE in the bottom 5km

of the troposphere. The cumulative distribution is con-

centrated around relatively large values of MSE: the 1st

percentile corresponds to an MSE of 333.8K, which

Fig. 6 indicates as the average value in the environment

at an altitude of 1.3 km. Notice that MSE is not con-

served under mixing, so our estimates are likely to be

biased low. Nevertheless, such low altitudes as those

indicated above support the claim that air in cold pool

cores is unlikely to originate in the midtroposphere.

Finally, the third method that we use to check the

consistency of our claim that initial heights of precipitation-

driven downdrafts are very low is based on the joint

distribution of MSE and vertical velocity at a given

height. In particular, Fig. 7 (left) shows the logarithm

of the time-averaged joint distribution of MSE

anomaly—computed at each time step with respect to

the horizontal average—and the vertical velocity at

531m near the top of the subcloud layer. The black

curve highlights the contour corresponding to the

value 4.5 of the logarithm of the distribution, which

encloses 99.7% of the total number of grid boxes

sampled and 87.5% of the total flux of MSE. The

contour line shows that most of the MSE flux is given

by grid boxes with an MSE anomaly higher than 26K

and relatively small vertical velocities, which suggests

that downdrafts from the midtroposphere do not inject

much air in the boundary layer.

FIG. 5. Time averages of the vertical profiles of rain evaporation

rate (blue), graupelmelting rate (green), and total precipitating water

specific humidity (red) for the bottom 5km of the model domain.

FIG. 6. Time average of the vertical profile of MSE in the envi-

ronment (blue) and cumulative distribution function of MSE in

cold pool cores (red).

FEBRUARY 2016 TORR I AND KUANG 845

b. On the contribution of precipitation-drivendowndrafts to the subcloud-layer MSE

At this point, it is natural to try and quantify the role

of the various transport categories—updrafts, down-

drafts, and the environment—in balancing the action of

surface fluxes in the subcloud layer. To begin with, let us

consider a column of air of unit surface and define the

mass contained in the subcloud-layer portion of the

column as

[r]5

ðztopzsfc

r dz , (2)

where ztop and zsfc are the heights of the top of the

subcloud layer and the surface. Because mass can be

exchanged between the subcloud layer and the free

troposphere, if we call Mi the mass flux associated with

the i transport category, the temporal variation in mass

per unit surface in the subcloud layer can then be written

as follows:

›[r]

›t5 �

i2TM

i, (3)

where T5 fupdraft, downdraft, environmentg. Along

the same lines of reasoning, the interaction of the sub-

cloud layer and the free troposphere causes the total

MSE per unit surface in the subcloud layer, defined by

[rh]5

ðztopzsfc

(rh) dz , (4)

to vary in time according to the following:

›[rh]

›t5 �

i2TM

ihi, (5)

where hi is the value of MSE at the top of the subcloud

layer associated with the transport category i. Notice

that the subcloud-layer MSE can also change because of

radiative effects, which we will represent as QR, and

through sensible and latent heat fluxes at the surface,

which we will write as SH and LH. It follows that the

complete form of Eq. (5) is given by

›[rh]

›t5 �

i2TM

ihi1 (SH1LH)1 [rQ

R] , (6)

where [rQR] is defined in a similar fashion to [rh]. If we

now assume that the air in the subcloud layer obeys the

Boussinesq approximation and has constant density r0,

we can rewrite the left-hand side of the above equation

as

›[rh]

›t5

ðztopzsfc

›r

›th1 r

›h

›tdz’

ðztopzsfc

�i2T

Mih dz1 r

0

›h

›tZ ,

(7)

FIG. 7. (left) Logarithm of the joint distribution of vertical velocity and MSE anomaly at 531m. The black

contour corresponds to the isopleth 21 of the logarithmic distribution. (right) Sinks of MSE at the top of the

subcloud layer divided by three transport categories: updraft, downdraft, and environment. The blue (green) bars

show the rates according to the broad (narrow) definition of the categories introduced in Thayer-Calder and

Randall (2015), whereas the red bars show the results obtained using the definition we have given using Lagrangian

particles.

846 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

where Z5 ztop 2 zsfc and h is the average MSE per unit

area in the subcloud layer. This can be further approx-

imated by substituting h in the integral with its value at

the top of the subcloud layer htop to give

›[rh]

›t’Z

�i2T

Mihtop

1 r0

›h

›t

!. (8)

Substituting Eq. (8) into Eq. (4), we can write a budget

equation for h:

›h

›t5

1

Z�i2T

wi(h

i2 h

top)1

SH1LH

r0Z

1QR, (9)

where wi is the velocity associated with the mass fluxMi

and we have substituted the term for the radiative ef-

fects with its subcloud-layer average QR. We have de-

rived this formula for an atmospheric column of unit

area, but a generalization to an extended domain such as

the one we are using is straightforward.

The choice of substituting h with htop in the right side

of Eq. (7) may have seemed a bit arbitrary, but it can be

fully appreciated in light of Eq. (9), where it is used as a

reference to determine the anomalies of MSE at the top

of the subcloud layer: both terms hi and htop are com-

puted at the same height.

Another source of arbitrariness that we have deliber-

ately left vague in our derivation is the definition of the

three transport categories. As is well known, the lines

between the three categories are blurry enough to allow

for multiple ways to address this issue. For example,

Thayer-Calder and Randall (2015) used two different

definitions to compute the budget ofMSEat the top of the

boundary layer. In the first one, called broad, each grid

box at 500-m height is considered a downdraft (updraft)

box if it contains any cloud ice, water, or precipitation and

has a negative (positive) velocity; if the box does not

contain any condensate, it is considered as environment.

In the second, called narrow, a grid box is considered a

downdraft (updraft) box if the vertical velocity is

below21ms21 (above 1ms21) for two continuous levels,

above and below 500m, and if it contains any condensate;

all other grid boxes are considered environment.

A potential problemwith this approach is that some of

the properties of a parcel, like its vertical velocity or its

liquid water specific humidity, near the top of the

boundary layer or cloud base are not necessarily in-

dicative of what transport category the parcel belongs

to. For instance, a parcel that will enter a convective

updraft might have an LCL at a higher altitude than

500m, or it might encounter significant convective in-

hibition, which will then sensibly reduce its vertical ve-

locity, before reaching its LFC. A similar story could

also hold true for parcels in downdrafts. We have tried

reproducing the results of Thayer-Calder and Randall

(2015) at various heights around 500m and found the

conclusions, particularly the contribution by updrafts, to

change according to which height is being used.

To remove this ambiguity, we propose a slightly dif-

ferent approach using the Lagrangian particles. Let us

go back to the joint distribution of vertical velocity and

MSE anomaly shown in Fig. 7 (left). Whenever a par-

ticle reaches the top of the subcloud layer, its vertical

velocity and the future 60min of its history are consid-

ered. If the particle has positive vertical velocity and,

within the specified time frame, acquires positive

buoyancy at an altitude higher than 1.5 km, the particle

is considered an updraft particle; if the particle has

negative vertical velocity and enters a grid box with a

density potential temperature anomaly lower than20.5K

in less than an hour, the particle is considered a downdraft

particle. If none of the above conditions are true, the

particle is considered an environment particle.

In Fig. 7 (right), we show the contributions to the budget

of MSE by the three transport categories. To compare our

method with those of Thayer-Calder and Randall (2015),

we measure the fluxes of MSE at 531m, which, in our

model, is the height closest to 500m and also corresponds

to the top of the subcloud layer. The blue (green) bars

refer to the results obtained using the broad (narrow)

definitions explained above, whereas the red bars refer to

the method using Lagrangian particles explained above.

Like Thayer-Calder and Randall (2015), we find turbulent

mixing in the environment to be the biggest contributor to

the total MSE sinks, although the contribution from up-

drafts and downdrafts is not negligible. The export of high

MSE air by convective updrafts provides a sink of MSE of

2.43Kday21, which is 16% of the total, amounting to

15.51Kday21, whereas the low-MSE air injected in the

subcloud layer by precipitation-driven downdrafts reduces

MSE at the rate of 1.81Kday21, 12% of the total. As

documented in the supplemental information, we have

also tested the sensitivity of the results to the sampling

height and found that our conclusions are robust.Of all the

heights where we have tested our results, 531m is the one

where the three methods are closest to each other, espe-

cially in predicting the contribution by downdrafts. Finally,

we want to note that the surface sensible and latent heat

fluxes contribute to the budget ofMSEwith a source equal

to 16.28Kday21, whereas radiative effects provide a sink

of 0.86Kday21.

c. On the maintenance of precipitation-drivendowndrafts

The next question that we want to focus on in the

present work is how to quantify the importance of rain

FEBRUARY 2016 TORR I AND KUANG 847

evaporation and whether it is bigger than that of con-

densate loading in maintaining precipitation-driven

downdrafts. Because we are concerned with the role of

rain evaporation from the dynamical point of view, we

propose using a purely dynamical quantity such as work

to address the problem.

The buoyancy of every particle can be defined in

terms of its density potential temperature:

b5 g

ur2 u

r

ur

!, (10)

where g is the gravitational constant and ur is the hori-

zontal average of density potential temperature.

At each time step, the contribution to a particle’s

buoyancy by the condensate loading bl is given by

bl52g

qlu

ur

!, (11)

from which it follows that the work done by condensate

loading on a particle in a precipitation-driven downdraft

can be defined as

Wl52g

ðzizcore

qlu

ur

!dz , (12)

where the integral is taken from the initial height zi to

the height where the particle enters the core of a cold

pool zcore.

To do the same for rain evaporation, we consider the

change in the thermodynamic state of a particle at each

time step due to the evaporation of a certain amount of

precipitation qev. Assuming that the evaporation takes

place at the end of each time step and that the MSE of a

particle is left invariant, the particle’s potential tem-

perature will change by

uev52

Ly

cp

qev, (13)

where Ly is the latent heat of vaporization, equal to

2.5104MJkg21 at 08C, and cp is the specific heat of dry

air at constant pressure, equal to 1004 J kg21K21. This

change in thermodynamic properties implies a change in

the particle’s buoyancy that, after a bit of algebra, can be

written as follows:

bev52

g

ur

"L

y

cp

(11 «qy2q

l)2 «u

#qev. (14)

Much in the same spirit of Eq. (12), we define the work

done on a particle by rain evaporation as

Wev52g

ðzizcore

"L

y

cp

(11 «qy2 q

l)2 «u

#qev

ur

dz . (15)

We will follow the convention that work done on a

particle by the environment is positive.

The definition we have given above will provide us

with a way to quantify the importance of each mecha-

nism in maintaining a downdraft. Considering again the

Lagrangian particles we sampled earlier to construct

Fig. 2, we compute the two types of work for each par-

ticle and average together the values for particles with

the same initial height with the assumption that they

have a comparable history and experienced the same

phenomena. Figure 8 shows a comparison of the average

work done on particles with different initial heights by

condensate loading (solid red) and by rain evaporation

(solid blue). Notice that, apart from the bottom 300m,

the contribution due to evaporation is smaller than that

due to loading by a factor of approximately 0.4.

We can test the consistency of this result by using an

Eulerian approach. First, we invoke some approxima-

tion to simplify Eqs. (12) and (15). To first order, we can

exchange ur with u in the denominator, and we can

consider qy and ql to be much less than 1. Then, since «u

is more than an order of magnitude smaller than the ratio

Ly/cp, we can neglect the second term in square brackets

in Eq. (15). With these simplifications, we can write

FIG. 8. Comparison between the total work done on Lagrangian

particles with different initial heights by different contributions to

buoyancy (solid lines) and total buoyancy (dashed lines). Line

colors refer to the contributions by condensate loading (red) and

the contribution by rain evaporation (blue).

848 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

Wev

Wl

’

L

y0

cpQ

!ð qevdzð

qldz

, (16)

where Ly has been approximated by its value at 08C,2.5104MJkg21, and Q represents a scale for the poten-

tial temperature and equals 300K. To evaluate the in-

tegrals in the above equation, we will consider a parcel

descending from an altitude of 1 km, the location of the

peak in Fig. 2. For every height, we compute the distri-

bution of negative vertical velocities of grid boxes where

qp is nonzero, and we assume that the vertical velocity of

the parcel we are considering equals the median of the

distribution. This construction is to ensure that the

parcel is representative of parcels within precipitation-

driven downdrafts.

The integral in the denominator is proportional to the

average value of liquid water content of the parcel, and

it can be quickly put in discrete form as

ðzizsfc

qldz5 �

ki

i5ksfc

(ql)iDz

i, (17)

whereDzi is the vertical grid spacing at height zi, and ksfcand ki are the model levels corresponding to heights zsfcand zi, respectively.

The integral in the numerator requires a little more

care. The integrand represents the specific humidity of

liquid water that is evaporated into the parcel as it

travels a distance equal to dz. By definition, this can be

rewritten as

qev5 (dqsrc

y ) dt5 (dqsrcy )

dz

w, (18)

where (dqsrcy ) is the rate of generation of water vapor

by rain evaporation and dt is the time it takes the

parcel to go from z1 dz to z. It follows that we can

evaluate the integral in the numerator of Eq. (16) by

computing the sum:

ðzizsfc

qevdz5 �

ki

i5ksfc

(dqsrcy )

i

�Dz2iw

i

�. (19)

The liquid water specific humidity and the evaporation

rate can be diagnosed from the model outputs, and their

values in the bottom 5km can be seen in Fig. 5. Plugging

in all the numbers, we can compute that the ratio be-

tween the work done by rain evaporation and that done

by condensate loading is 0.36, which compares reason-

ably well with the estimate of 0.41 that is obtained using

the Lagrangian particles.

As discussed by Davies-Jones (2003), buoyancy in a

nonhomogeneous environment, such as the one we are

examining, is an ill-defined concept. Instead, what

should really be considered as the effective buoyancy

experienced by a parcel is the sum of the Archimedean

buoyancy, which we have defined in Eq. (1), and the

buoyancy-related pressure gradient, which cannot be

determined locally. Recently, it has been shown (Torri

et al. 2015; Jeevanjee and Romps 2015) that parcels that

have been triggered by cold pools and that are ascending

toward their LFC are subject to a significant cancellation

between buoyancy and buoyancy-related pressure gra-

dients. This leads us to ask whether our conclusions on

the minor role played by rain evaporation in maintain-

ing the downdraft are robust when buoyancy-related

pressure gradients are taken into account. To address

this point, for simplicity, we first assume that the con-

tributions to the buoyancy-related pressure gradients at

every moment can be viewed as approximately pro-

portional to the contributions to the Archimedean

buoyancy by a factor given by the ratio of the buoyancy-

related pressure gradient and buoyancy, which we will

refer to as buoyancy ratio b:

b 5def ›

zpb

b’›zpb,l

bl

’›zpb,ev

bev

, (20)

where pb is the buoyancy-related pressure perturbation

and pb,l and pb,ev are its contributions due to condensate

loading and rain evaporation, respectively.

In this way, we can reconstruct the various contribu-

tions to the buoyancy-related pressure gradients very

simply, and the calculation of the work done by these

contributions is straightforward.

The two contributions to the work done by total

buoyancy—bywhichwemean the sumof theArchimedean

buoyancy and the buoyancy-related pressure gradients

(Torri et al. 2015)—are shown by the dashed lines in

Fig. 8, and they suggest that our conclusions are robust,

even when buoyancy-related pressure gradients are

taken into account. A close inspection of the vertical

profiles of buoyancy ratio (not shown) suggests that this

quantity tends to be maximum in magnitude in the

subcloud layer, with values approaching 21, and be-

comes closer to 0 with increasing height. This explains

why the particles that seem most affected by the cor-

rection are those with an initial height smaller than 1km.

A natural question that arises now is whether con-

densate loading dominates over rain evaporation at any

moment during the particle’s descent in the downdraft

or if the two mechanisms are important at different

stages. To investigate this, instead of considering an in-

tegrated quantity such as the work, we look directly at

FEBRUARY 2016 TORR I AND KUANG 849

the contributions to buoyancy given by the two mecha-

nisms. We present the results in Fig. 9, with Fig. 9 (left)

referring to the contribution from condensate loading

and Fig. 9 (right) referring to that from rain evaporation.

As for the construction of Fig. 8, we use the initial

height of a particle as a sampling criterion: the histories

of particles with the same initial height are averaged

together, and the results are presented for each value of

initial height. Thus, each column in Fig. 9 represents the

vertical profile of the average contribution to the

buoyancy of particles having an initial height given by

the abscissa of the column in the figure.

In comparing the two plots, one should be careful to

notice that the color bars differ by a factor of 2.5. This

was done to make it possible for the eye to easily dis-

tinguish the main features of each contribution at every

height for different types of particles. Therefore, in spite

of the immediate appearance, condensate loading con-

tributes to buoyancy more than evaporation almost ev-

erywhere, except for very near the surface, where both

contributions seem of equal magnitude.

Another interesting feature of the two plots is that, for

any initial height, the altitude where each contribution is

maximum is different: for loading, the peak tends to be

in the initial stages of the downdraft, whereas for

evaporation the maximum is in the subcloud layer.

For the condensate loading, the peak is essentially due

to two factors: the first is that rain columns are not steady

but, rather, quickly grow to a maximum and then decay

over time; the second is the fact that precipitating water

in a rain column flows through the Lagrangian particles

or, in other words, the vertical velocity of the particles is

typically smaller than the terminal velocity of raindrops.

To check that the latter is true, we can consider

Fig. 10, which shows the average vertical velocity

experienced by particles of various initial heights at

different stages of their descent. Above the bottom few

hundred meters, where the hydrostatic pressure per-

turbation in the core of cold pools causes the downdraft

to slow down, the vertical velocities experienced by

the descending particles are roughly between 20.5

FIG. 9. Vertical profiles of the time-averaged change of buoyancy due to (left) condensate loading and (right) rain

evaporation for Lagrangian particles in precipitation-driven downdrafts with different initial heights.

FIG. 10. Vertical profiles of time-averaged vertical velocities for

Lagrangian particles in precipitation-driven downdrafts with dif-

ferent initial heights.

850 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

and 21.5m s21. Considering the precipitation rate and

the precipitating liquid water content in the grid boxes

of the bottom 3km of the model, we can estimate that

the average terminal velocities of raindrops are between

25 and 26ms21, well outside the range of average ve-

locities of Lagrangian particles within precipitation-driven

downdrafts. As a side note, we want to point out that the

terminal velocities diagnosed from the model are consis-

tent with data from observations: the terminal velocities

for raindrops with diameters between 1 and 5mm go

from 24.03 to 29.09m s 21 (Gunn and Kinzer 1949).

To understand why the contribution to buoyancy by

rain evaporation—and, more generally, the evaporation

rate—peaks at low altitudes, we must take a step back

and think about how liquid water evaporates in a par-

ticle in the downdraft. It is well known (see, e.g.,

Kamburova and Ludlam 1966; Betts and Silva Dias

1979) that evaporation of a falling raindrop of radius

r and mass m can be modeled as a diffusion process:

Dm

Dt5 4pDC

yrr(q

y2 q

sat) , (21)

whereD is the diffusion coefficient of water vapor in air,

Cy is a ventilation factor, r is the density of the parcel,

and qsat is the saturation specific humidity. If we consider

an air parcel with volume V and assume, for simplicity,

that it contains N raindrops of equal radius R, then the

liquid water content of the parcel can be simply related

to the radius of the raindrop by the following:

ql5

�4pr

lN

3rV

�R3 5

def�4pDC

y

VK

�3

R3 , (22)

where rl is the density of liquid water, and, for simplicity,

we have defined a new variable K as

K5DCy

�48p2r

NV2rl

�1/3

. (23)

With some simple algebra, it is easy to show that Eq. (21)

can be rewritten as

Dql

Dt5Kq1/3

l (qy2 q

sat) . (24)

Figure 11 shows the average product of the cubic root of

the cloud liquid water and the saturation deficit for

particles with different initial heights. Assuming that K

is constant at all heights and for particles from all initial

heights, the figure can be compared directly with the

contribution to buoyancy by rain evaporation. Notice

that, for each initial height, the profiles show a peak in

the subcloud layer, roughly at the same height where the

contribution by evaporation is maximum. Thus, one

could say that the peak in evaporation rate in the sub-

cloud layer is the result of two factors: the saturation

deficit, which increases as the surface is approached (not

shown), and the liquid water content of each parcel,

which, as can be inferred from Fig. 9 (left), decreases in

the latter stages of the downdraft.

Given the crude assumptions that we have made to

derive Eq. (24), it is not surprising that the agreement

between Fig. 11 and Fig. 9 (right) is only partial: unlike

the profiles of the contribution by evaporation, the

magnitude of the product seems generally to decrease

for particles with increasing initial height. We expect

that accounting for a distribution of radii and by allow-

ing the number of raindrops N to vary with height and

initial height would improve the agreement. However,

we want to stress that this mismatch does not change our

conclusions.

4. Conclusions

We have used an LPDM to address two important

aspects regarding the dynamics of precipitation-driven

downdrafts: namely, their initial height and the role

played by rain evaporation in maintaining them. By

sampling all the Lagrangian particles that are injected in

the coldest regions of cold pools in the subcloud layer by

downdrafts, we have determined that the vast majority

of such particles start descending from a height lower

FIG. 11. Profiles of the time average of the cubic root of liquid

water specific humidity multiplied by the saturation deficit for

Lagrangian particles in precipitation-driven downdrafts.

FEBRUARY 2016 TORR I AND KUANG 851

than 2.5 km in the model domain. Looking at the posi-

tion of the particles at times prior to their entrance in a

precipitation-driven downdraft, we have shown that

most of them originate in the subcloud layer; almost

none come from high altitudes.

As a further check, we have broadened our sampling

criterion and considered all particles that are part of

downdrafts at any height, regardless of whether they will

be injected in a cold pool or not. With this, we have

shown that downdrafts exist at all heights in the

model but a few hundred meters past their initial

height. This provided additional support to the claim

that precipitation-driven downdrafts originate at very

low altitudes and do not, therefore, bring very cold and

dry air into the subcloud layer.

This led us to the question of what balances the sur-

face fluxes and keeps the boundary layerMSE relatively

steady in time. Using a definition of the three transport

categories—updraft, downdraft, and environment—

based on the Lagrangian particles, we have concluded

that most of theMSE sink is to be attributed to turbulent

mixing or wave activity in the environment, although the

contributions by updrafts and downdrafts are non-

negligible, their sum accounting for 28% of the total.

We then proceeded to quantify the role of rain

evaporation in maintaining the precipitation-driven

downdrafts. By looking at the total work done on each

Lagrangian particle that is injected in the core of a cold

pool by a downdraft and distinguishing the contribution

due to condensate loading from that due to rain evap-

oration, we showed that the latter is smaller than the

former, roughly by a factor of 0.4.

Considering the change of the Lagrangian particles’

buoyancy due to condensate loading and rain evapora-

tion, we showed that the former is bigger than the latter

at all heights and tends to bemaximum in the first part of

the downdraft descent; on the other hand, the latter is

highest in the second part of the descent, when the

particles are in the subcloud layer.

We interpreted the first effect as due to the fact that

the terminal velocity of rain droplets is higher in mag-

nitude than the vertical velocity of the descending

Lagrangian particles and that the liquid water content as-

sociated with each rain shaft is not constant in time.

As for rain evaporation, we have argued that its

maximum in the subcloud layer is simply the combination

of saturation deficit and the liquid water specific humidity,

which enters the equation for evaporation rates only in a

cubic root. In particular, even though the precipitation

experienced byLagrangian particles is largest in the initial

stages of the downdraft, the small saturation deficit does

not allow for large amounts of rain to be evaporated.Only

in the subcloud layer is the saturation deficit large enough

to make the contribution of rain evaporation in main-

taining the downdraft more significant.

Since the dynamics of downdraft maintenance is

tightly related to microphysical processes such as rain

evaporation, it seems natural to ask how sensitive our

conclusions are to the microphysics scheme. For this

reason, we have run the same simulation described in

section 2 using the two-moment microphysics scheme

introduced inMorrison et al. (2005) and have carried out

the same analysis described in the previous section.

Apart from some slight differences, we found that the

conclusions of this study are robust.

Although this robustness gives us some confidence

over the generality of our findings, we want to stress that

we have only considered an oceanic case in RCEwith no

wind shear and, a priori, we do not expect our results to

hold for other cases. For instance, it is very well possible

that, in a continental case with a much drier boundary

layer, evaporation of rain will play a much bigger role in

maintaining the downdraft. This could also be the case in

the presence of wind: a considerable vertical wind shear

might change the geometry of a precipitating cloud,

exposing a larger part of a precipitating column to an

unsaturated environment.

Finally, the conceptual framework that emerges

from this manuscript can have important consequences

for climate models. In particular, our findings suggest

that the premises on which the BLQE hypothesis is

founded—that downdrafts are responsible for most of

the MSE sink in the boundary layer—may not be real-

ized in nature. Although some models have been shown

to better reproduce certain features of the tropical cli-

mate, such as the MJO, when strong convective down-

drafts are included (Mishra and Sahany 2011), we suggest

that the key for amore accurate simulation of the tropical

atmosphere is in improving parameterizations of the

transport due to turbulent mixing or wave activity in the

environment. Mesoscale organization of convective sys-

tems, absent in this study, may also affect the results.

Notwithstanding these additional scenarios, which clearly

warrant further investigations, our analysis of the oceanic

RCE case addresses a basic scenario that is potentially

representative of large areas over the tropical oceans with

modest wind shears.

Although they play only a secondary role in the MSE

budget of the tropical boundary layer, downdrafts still

remain an important component of convective systems

that should be included in convective parameterizations.

Our results could also be useful for this task: first, they

indicate the initial height tends to be in the lower tro-

posphere; second, the fact that rain evaporation contrib-

utesmuch less than condensate loading inmaintaining the

downdraft indicates that downdrafts might be more

852 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 73

sensitive to the model precipitation rate rather than to the

relative humidity.

Acknowledgments. The authors thank Marat

Khairoutdinov for access to the SAM code and Paul

Edmon for precious and tireless assistance with the

HarvardOdyssey cluster on which the simulations were

run. This research was partially supported by the Office

of Biological and Environmental Research of the U.S.

Department of Energy under Grant DE-SC0008679 as

part of the ASR program and by NSF Grants AGS-

1062016 and AGS-1260380.

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